In the case in which SHDM interacts with SM particles only through the gravitational interaction, assuming the very general case of a DS characterised by its own non-abelian gauge symmetry, SHDM can decay only through non-perturbative effects as the instanton-induced decay (see [1]} and references therein). In this case the lifetime of SHDM follows from the corresponding instanton transition amplitude that, being exponentially suppressed, provides long-living particles. Considering the zeroth oder contribution, the instanton-induced lifetime can be written as \(\tau _X\simeq M_X^{-1} \exp {(4\pi /\alpha _X)}\) , being \(\alpha _X\)  the reduced coupling constant of the hidden gauge interaction in the DS.
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